Effects of slip condition on MHD stagnation-point flow over a power-law stretching sheet

The steady two-dimensional magnetohydrodynamic stagnation flow towards a nonlinear stretching surface is studied. The no-slip condition on the solid boundary is replaced with a partial slip condition. A scaling group transformation is used to get the invariants. Using the invariants, a third-order ordinary differential equation corresponding to the momentum is obtained. An analytical solution is obtained in a series form using a homotopy analysis method. Reliability and efficiency of series solutions are shown by the good agreement with numerical results presented in the literature. The effects of the slip parameter, the magnetic field parameter, the velocity ratio parameter, the suction velocity parameter, and the power law exponent on the flow are investigated. The results show that the velocity and shear stress profiles are greatly influenced by these parameters.     

Oblique stagnation slip flow of a micropolar fluid

This paper considers the problem of steady two-dimensional boundary layer flow of a micropolar fluid near an oblique stagnation point on a fixed surface with Navier’s slip condition. It is shown that the governing nonlinear partial differential equations admit similarity solutions. The resulting nonlinear ordinary differential equations are solved numerically using the Keller box method for some values of the governing parameters. It is found that the flow characteristics depend strongly on the micropolar and slip parameters.     

The effect of the slip condition on unsteady flow due to non-coaxial rotations of disk and a fluid at infinity

An exact solution of the Navier-Stokes equation is constructed for the magnetohydrodynamic (MHD) flow. The flow is due to non-coaxially rotations of a porous disk with slip condition and a fluid at infinity. The solutions for steady and unsteady cases are obtained by Laplace transform method. The effects of magnetic field and slip parameters are shown and discussed.     

二阶滑移边界对微型气浮轴承稳态性能的影响

考虑微型气浮轴承的尺寸特征，内部气流不再满足连续流的假设，根据Knudsen数可确定内 部气流为滑移流. 分别利用一阶速度滑移模型和二阶速度滑移模型对连续流的状态方程进行 修改，得到一阶滑移流和二阶滑移流机制下修正的雷诺方程. 利用有限差分法对连续流、一 阶滑移流和二阶滑移流的雷诺方程分别求解，得出相应的承载力和偏位角. 经过对比分析， 发现采用滑移流模型得到的轴承的稳态力学性能与连续流机制的结果存在较大差异，一阶滑 移流与二阶滑移流间的差异随偏心率增加而增加. 说明在MEMS环境下必须考虑滑移流效应 对微型气浮轴承稳态力学性能的影响. 在大偏心率工作状态下，二阶滑移流模型能够得到最 好的结果.     

Continuum analytical modelling of thermal creep

A thermal creep process is studied in quite wide rectangular micro channels, sufficiently wide so that it is possible to consider this configuration as two parallel plates. The inlet and outlet reservoirs are maintained at the same constant pressure. A constant temperature gradient exists along the walls of the channel joining the two tanks. Thus a gas flow is induced and thermally sustained until steady conditions are reached. A complete analytical solution is derived in slip regime, yielding all the flow parameters, for Knudsen numbers smaller than 0.25. The analytical results are in good agreement with the numerical “exact” solution of the continuum equation system. Furthermore our continuum approach data are compared to those deduced from approaches based on Boltzmann equation model treatments: these various methods lead generally to a satisfactory agreement between their respective mean parameters. Nevertheless significant differences appear on the transversal velocity profiles and are further discussed.     

One dimensional numerical simulation for steady annular condensation flow in rectangular microchannels

A one dimensional model for steady annular condensation flow in rectangular microchannels is developed and numerically solved under constant heat flux condition. The results indicate that the annular condensation length is determined by the contact angle, heat flux, vapor pressure, hydraulic diameter and aspect ratio of rectangular microchannels. A larger inlet vapor pressure and hydraulic diameter or a smaller heat flux and contact angle can all result in a longer annular condensation length. In addition, the simulation results of steady annular condensation flow in rectangular microchannels are compared with that in triangular microchannels. The differences in curvature radius, condensate pressure and velocity, vapor velocity distributions in rectangular and triangular microchannels under the same conditions verify the considerable influence of cross-section shape on micro flow condensation.     

Ion slip effect on a steady flow through a circular pipe of a dusty conducting Oldroyd 8-constant fluid

In this paper, a steady magnetohydrodynamic (MHD) flow of a dusty incompressible electrically conducting Oldroyd 8-constant fluid through a circular pipe is examined with considering the ion slip effect. A constant pressure gradient in the axial direction and an external uniform magnetic field in the perpendicular direction are applied. A numerical solution is obtained for the governing nonlinear momentum equations by using finite differences. The effect of the ion slip, the non-Newtonian fluid characteristics, and the particle-phase viscosity on the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is reported.     

Flow and heat transfer of an electrically conducting third grade fluid past an infinite plate with partial slip

The effects of partial slip on the steady flow and heat transfer of an electrically conducting, incompressible, third grade fluid past a horizontal plate subject to uniform suction and blowing is investigated. Two distinct heat transfer problems are studied. In the first case, the plate is assumed to be at a higher temperature than the fluid; and in the second case, the plate is assumed to be insulated. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. Numerical solutions for the governing nonlinear equations are obtained over the entire range of physical parameters. The effects of slip, magnetic parameter, non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. It is interesting to find that the velocity and the thermal boundary layers decrease with an increase in the slip, and as the slip increases to infinity, the flow behaves as though it were inviscid.     

坐标变换法数值求解微通道行波电场电渗流

采用坐标变换法数值求解了耦合的Poisson-Nernst-Planck (PNP)方程和Navier-Stokes(NS)方程, 研究二维狭窄微通道行波电场电渗流数值解. 数值结果表明,坐标变换法能有效降低电渗流解数值解在双电层的高梯度, 有效改善数值解的收敛性和稳定性. 坐标变换的电渗流数值解和原始坐标下的数值解完全一致. 坐标变换后采用简单的网格也能得到和原始坐标下复杂网格相同的解. 给出了滑移边界的近似解与完整的PNP-NS数值解的比较. 在双电层厚度与微通道深度比值(λ/H)很小的情况下(相对深通道), 两者的解基本一致. 但在λ/H较大时(相对浅通道)滑移边界的解高于电渗流速度.      

Relaxation effects of slip in shear flow of linear molten polymers

The transient shear response of a linear molten polymer (linear low-density polyethylene) in the nonlinear domain was studied using a true shear (sliding plate) rheometer with different gap spacings to detect slip effects. It was found that nonlinear viscoelasticity is further complicated by wall slip phenomena. Experimental evidence suggested that static slip models coupled with Wagner’s constitutive equation cannot adequately describe the experimental data at large and fast shear deformations. A new dynamic slip model involving multiple slip relaxation times is proposed in this paper, together with a method to assess the model parameters. Significant improvement in predicting the stress response is demonstrated by several examples of start-up of steady shear and large-amplitude oscillatory tests of a linear low-density polyethylene.     

Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane

A thin film of a power–law fluid flowing down a porous inclined plane is considered. It is assumed that the flow through the porous medium is governed by the modified Darcy’s law together with Beavers–Joseph boundary condition for a general power–law fluid. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium and a slip condition at the bottom is used to incorporate the effects of the permeability of the porous substrate. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. The results show that the substrate porosity in general destabilizes the film flow system and the shear-thinning rheology enhances this destabilizing effect. A weakly nonlinear stability analysis reveals the existence of supercritical stable and subcritical unstable regions in the wave number versus Reynolds number parameter space. The numerical solution of the nonlinear evolution equation in a periodic domain shows that the fully developed nonlinear solutions are either time-dependent modes that oscillate slightly in the amplitude or time independent stable two-dimensional nonlinear waves with large amplitude referred to as ‘permanent waves’. The results show that the shape and the amplitude of the nonlinear waves are strongly influenced by the permeability of the porous medium and the shear-thinning rheology.     

Mass flow rate measurements in gas micro flows

The main objective of this experimental investigation on the gas flow slip regime is to measure the mass flow rate in isothermal steady flows through cylindrical micro tubes. Two technical procedures devoted to mass flow rate measurements are compared, and the measured values are also compared with the results yielded by different approximated analytical solutions of the gas dynamics continuum equations. Satisfactory results are obtained and the way is clearly opened to measuring mass flow rates for higher Knudsen numbers, over all the micro flow transitional regime.     

REDUCED-ORDER MODELS OF UNSTEADY TRANSONIC VISCOUS FLOWS IN TURBOMACHINERY

The proper orthogonal decomposition (POD) technique is applied in the frequency domain to obtain a reduced-order model of the unsteady flow in a transonic turbomachinery cascade of oscillating blades. The flow is described by a inviscid—viscous model, i.e. a full potential equation outer flow model and an integral equation boundary layer model. The nonlinear transonic steady flow is computed first and then the unsteady flow is determined by a small perturbation linearization about the nonlinear steady solution. Solutions are determined for a full range of frequencies and validated. The full model results and the POD method are used to construct a reduced-order model in the frequency domain. A cascade of airfoils forming the Tenth Standard Configuration is investigated to show that the reduced-order model with only 15–75 degrees of freedom accurately predicts the unsteady response of the full system with approximately 15 000 degrees of freedom.     

Numerical modelling of wind flow over buildings in two dimensions

Numerical solutions to the Navier–Stokes equation may provide designers with predictions of the wind environment of buildings under design. To investigate this possibility, two complementary solution procedures are implemented for two-dimensional geometry: a random vortex method to depict the flow evolution, and a control volume method to depict the steady flow field. These are both illustrated by specific application to the case of a building form with a roof of arbitrary pitch.     

Influence of wall slip on the hydrodynamic behavior of a two-dimensional slider bearing

In the present paper, a multi-linearity method is used to address the nonlinear slip control equation for the hydrodynamic analysis of a two-dimensional (2-D) slip gap flow. Numerical analysis of a finite length slider bearing with wall slip shows that the surface limiting shear stress exerts complicated influences on the hydrodynamic behavior of the gap flow. If the slip occurs at either the stationary surface or the moving surface (especially at the stationary surface), there is a transition point in the initial limiting shear stress for the proportional coefficient to affect the hydrodynamic load support in two opposite ways: it increases the hydrodynamic load support at higher initial limiting shear stresses, but decreases the hydrodynamic load support at lower initial limiting shear stresses. If the slip occurs at the moving surface only, no fluid pressure is generated in the case of null initial limiting shear stress. If the slip occurs at both the surfaces with the same slip property, the hydrodynamic load support goes off after a critical sliding speed is reached. A small initial limiting shear stress and a small proportionality coefficient always give rise to a low friction drag. The project supported by the National Natural Science Foundation of China (10421002, 10332010), the National Basic Research Program of China (2006CB601205), and the Science Research Foundation of Liaoning Province (20052178). The English text was polished by Yunming Chen.     

Navier-Stokes方程二阶速度滑移边界条件的检验

对微尺度气体流动,Navier-Stokes方程和一阶速度滑移边界条件的结果与实验数据相比,在滑移区相互符合,在过渡领域则显著偏离.为改善Navier-Stokes方程在过渡领域的表现,有些研究者尝试引入二阶速度滑移边界条件,如Cercignani模型,Deissler模型和Beskok-Karniadakis模型.以微槽道气体流动为例,将Navier-Stokes方程在不同的二阶速度滑移模型下的结果与动理论的直接模拟Monte Carlo(DSMC)方法和信息保存(IP)方法以及实验数据进行比较.在所考察的3种具有代表性的二阶速度滑移模型中,Cercignani模型表现最好,其所给出的质量流率在Knudsen数为0.4时仍与DSMC和IP结果相符;然而,细致比较表明,Cercignani模型给出的物面滑移速度及其附近的速度分布在滑流区和过渡领域的分界处(Kn=0.1)已明显偏离DSMC和IP的结果.     

Efficient ADI and spline ADI methods for the steady-state Navier-Stokes equations

The present paper provides an improved alternating direction implicit (ADI) technique as well as high-order-accurate spline ADI method for the numerical solution of steady two-dimensional incompressible viscous flow problems. The vorticity-stream function Navier-Stokes equations are considered in a general curvilinear coordinate system, which maps an arbitrary two-dimensional flow domain in the physical plane into a rectangle in the computational plane. The stream function equation is parabolized in time by means of a relaxation-like time derivative and the steady state solution is obtained by a time-marching ADI method requiring to solve only 2 × 2 block-tridiagonal linear systems. The difference equations are written in incremental form; upwind differences are used for the incremental variables, for stability, whereas central differences approximate the non-incremental terms, for accuracy, so that, at convergence, the solution is free of numerical viscosity and second-order accurate. The high-order-accurate spline ADI technique proceeds in the same manner; in addition, at the end of each two-sweep ADI cycle, the solution is corrected by means of a fifth-order spline interpolating polynomial along each row and column of the computational grid, explicitly. The validity and the efficiency of the present methods are demonstrated by means of three test problems.     

Effect of Viscous Dissipation on the Boundary Layer Flow and Heat Transfer Past a Permeable Stretching Surface Embedded in a Porous Medium with a Second-Order Slip Using Chebyshev Finite Difference Method

This article investigates a theoretical and numerical study for the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid toward a permeable stretching surface embedded in a porous medium with a second-order slip and thermal slip. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting ODEs are successfully solved numerically with the help of Chebyshev finite difference method. Graphically results are shown for non-dimensional velocities and temperature. The effects of the porous parameter, the suction (injection) parameter, Eckert number, first- and second-order velocity slip parameter, the thermal slip parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. A comparison of numerical results is made with the earlier published results under limiting cases.     

复杂微通道气体流动的格子Boltzmann模拟

构建了一个模拟复杂微通道内气体流动的多松弛格子Boltzmann模型。该模型采用动力学曲面滑移边界,考虑了微尺度效应和努森层影响。此外,为了更准确地描述微通道内气体的滑移速度,在模型中引入孔隙局部Kn数来代替平均Kn数。之后采用Poiseuille流对模型进行验证,模拟结果与用直接模拟蒙特卡洛方法和分子模拟结果吻合较好,证明了该模型模拟微通道内处于滑移区和过渡区气体流动的有效性。最后,采用该模型模拟多孔介质内气体渗流过程。结果表明,随着孔隙平均Kn数的增加,多孔介质内的高渗区域增加,且优先从小孔隙中开始增加,这是由于小孔隙中微尺度效应更加明显,相对大孔隙流动阻力更小所致。     

On Nonlinear Evolution Approach for Convective Flow during Alloy Solidification

We consider the problem of nonlinear buoyant flow in a horizontal mushy layer during alloy solidification. We study the nonlinear evolution of such flow based on a recently developed realistic model for the mushy layer. The evolution approach is based on a Landau type equation for the amplitude of the secondary nonlinear solution, which is derived in this article. Using both analytical and computational methods, we calculate the solution to the evolution equation for both subcritical and supercritical regimes. We find, in particular, that for a passive mush, where the permeability is constant, and supercritical regime, the primary solution is linearly unstable to the secondary solution which becomes a steady stable solution for sufficiently large time, while the secondary solution decays to zero for the subcritical regime. On the other hand, for a realistic reactive mush, where the permeability is variable, the secondary flow can break down in a finite time for either supercritical or subcritical regime, which indicates existence of some kind of bursting behavior. These results are then compared to the corresponding ones based on the weakly nonlinear theory.